Nonhomogeneous Second Order ODE containing log

In summary, the person is asking for help with an ODE containing logs. They have tried solving it using the method of undetermined coefficients, but are unsure if their solution is correct. They are advised to try the operator method or the method of variation of parameters. They are also directed to a tutorial and a forum where they can get step-by-step help with their problem.
  • #1
crowy
1
0
Hi guy,

I have this ODE that I'm having problems with

y"+4y'+4y= e^(-2x)logx

Now, Using method of UC to get rid of the RHS I've tried using Ae^(-2x) x^2 logx

However, I'm not quite sure whether that is correct or not as I have never had a question containing logs before
 
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  • #2
If it is not a must that you have to use method of undetermined coefficients, you can have a look of the operator method. In this case, you don't have to care about what kind of non-homogeneous function you have. At least you can write the solution in integral form. Please refer to my tutorial in http://www.voofie.com/concept/Mathematics/" :

http://www.voofie.com/content/6/introduction-to-differential-equation-and-solving-linear-differential-equations-using-operator-metho/"

For equations with variable coefficients, you can at least try to find the solution using the below method:

http://www.voofie.com/content/84/solving-linear-non-homogeneous-ordinary-differential-equation-with-variable-coefficients-with-operat/"

If you want to see how to solve it step by step, you can try to ask it in http://www.voofie.com/concept/Mathematics/" by submitting a new question, and I am willing to solve it for you.
 
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  • #3
See the attachement :
 

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  • #4
In general, the "method of undetermined coefficients" can only be used when the right hand side is the type of function that might be a solution to a homogeneous linear equation with constant coefficients- sine and cosine, polynomials, exponentials, and combinations of those.

The method of "variation of parameters" works with any functions- although it may result in integrals the require non-elementary functions.
 
  • #5
.

Hi there,

It looks like you are on the right track with your approach using the method of undetermined coefficients. However, in order to solve this nonhomogeneous second order ODE, you will need to also consider the homogeneous solution (y_h) and the particular solution (y_p). The homogeneous solution can be found by setting the right hand side equal to zero and solving the resulting homogeneous differential equation. The particular solution can then be found using the method of undetermined coefficients, where you will need to consider the form of the particular solution to be Ae^(-2x) x^2 logx + Be^(-2x) x logx + Ce^(-2x) logx + De^(-2x). I hope this helps and good luck with your problem! Remember to always check your solution by plugging it back into the original equation.
 

1. How do you solve a nonhomogeneous second order ODE containing a logarithmic term?

To solve this type of ODE, we can use the method of undetermined coefficients or variation of parameters. Both methods involve finding a particular solution that satisfies the nonhomogeneous term, and then adding it to the general solution of the corresponding homogeneous ODE.

2. What is the difference between a homogeneous and nonhomogeneous ODE?

A homogeneous ODE is one in which all terms contain the dependent variable and its derivatives, while a nonhomogeneous ODE has additional terms that do not contain the dependent variable or its derivatives. Nonhomogeneous ODEs are typically more difficult to solve, as they require finding a particular solution in addition to the general solution.

3. Can a logarithmic term in an ODE be considered as a constant?

No, a logarithmic term cannot be considered as a constant in an ODE. It is a function that depends on the independent variable, and thus cannot be treated as a constant when solving the ODE.

4. How do you know if a particular solution is valid for a nonhomogeneous ODE with a logarithmic term?

To determine the validity of a particular solution, we can substitute it into the nonhomogeneous ODE and check if it satisfies the equation. If it does, then the particular solution is valid. If not, then we need to adjust the form of the particular solution and try again.

5. Can a nonhomogeneous ODE with a logarithmic term have more than one particular solution?

Yes, a nonhomogeneous ODE with a logarithmic term can have more than one particular solution. This is because the method of undetermined coefficients can sometimes yield multiple solutions, and the correct particular solution needs to be determined through substitution and verification.

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